Tangent vector of a curve

Question

Let $\vec{\sigma}:[a,b]\longrightarrow \mathbb{R}^2$ be a regular and closed curve of class $C^1$, parametrized respect to the arc lenght. Is true that the map $\vec{\sigma}':[a,b]\longrightarrow S^1$ is surjective?

Answer 1

This is not true.

Note that in the above drawing your map never reaches $(1,0)$.